0.06e: Exercises - Rational Expressions

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- 0.06: Review - Rational Expressions
- 0.07: Review - Solving Linear Equations

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A: Simplify Rational Expressions.

Exercise $\PageIndex{1}$

$\bigstar$ Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.

$\dfrac { 25 x ^ { 9 } } { 5 x ^ { 5 } }\\[6pt]$
$\dfrac { 64 x ^ { 8 } } { 16 x ^ { 3 } }\\[6pt]$
$\dfrac { x ^ { 2 } - 64 } { x ^ { 2 } + 16 x + 64 }\\[6pt]$
$\dfrac { x ^ { 2 } + x - 20 } { x ^ { 2 } - 25 }\\[6pt]$
$\dfrac { 9 - 4 x ^ { 2 } } { 2 x ^ { 2 } - 5 x + 3 }\\[6pt]$

$\dfrac { x - 3 x ^ { 2 } } { 9 x ^ { 2 } - 6 x + 1 }\\[6pt]$
$\dfrac { 2 x ^ { 2 } - 8 x - 42 } { 2 x ^ { 2 } + 5 x - 3 }\\[6pt]$
$\dfrac { 6 x ^ { 2 } + 5 x - 4 } { 3 x ^ { 2 } + x - 4 }\\[6pt]$
$\dfrac { x ^ { 3 } + x ^ { 2 } - x - 1 } { x ^ { 2 } + 2 x + 1 }\\[6pt]$
$\dfrac { 2 x ^ { 3 } - 5 x ^ { 2 } - 8 x + 20 } { 2 x ^ { 2 } - 9 x + 10 }\\[6pt]$

$\dfrac { 66 x ( 2 x - 5 ) } { 18 x ^ { 3 } ( 2 x - 5 ) ^ { 2 } }\\[6pt]$
$\dfrac { 26 x ^ { 4 } ( 5 x + 2 ) ^ { 3 } } { 20 x ^ { 5 } ( 5 x + 2 ) }\\[6pt]$
$\dfrac { x ^ { 2 } + 5 x + 6 } { x ^ { 2 } - 5 x - 14 }\\[6pt]$
$\dfrac { x ^ { 2 } - 8 x + 12 } { x ^ { 2 } - 2 x - 24 }\\[6pt]$
$\dfrac { 1 - x ^ { 2 } } { 5 x ^ { 2 } + x - 6 }\\[6pt]$

$\dfrac { 4 - 9 x ^ { 2 } } { 3 x ^ { 2 } - 8 x + 4 }\\[6pt]$
$\dfrac { 4 x ^ { 2 } + 15 x + 9 } { 9 - x ^ { 2 } }\\[6pt]$
$\dfrac { 6 x ^ { 2 } + 13 x - 5 } { 25 - 4 x ^ { 2 } }\\[6pt]$
$\dfrac { x ^ { 2 } - 5 x + 4 } { x ^ { 3 } - x ^ { 2 } - 16 x + 16 }\\[6pt]$
$\dfrac { x ^ { 4 } + 4 x ^ { 2 } } { x ^ { 3 } + 3 x ^ { 2 } + 4 x + 12 }\\[6pt]$

Answers to odd exercises:

$5 x ^ { 4 }; x \neq 0 \\[6pt]$
3.

$\dfrac { x - 8 } { x + 8 }; x \neq -8 \\[6pt]$
5.

$- \dfrac { 2 x + 3 } { x - 1 } ; x \neq 1, \dfrac { 3 } { 2 } \\[6pt]$
7.

$\dfrac { 2 ( x - 7 ) } { 2 x - 1 }; x \neq -3, \dfrac { 1 } { 2 } \\[6pt]$

$x - 1; x \neq -1 \\[6pt]$
11.

$\dfrac { 11 } { 3 x ^ { 2 } ( 2 x - 5 ) } ; x \neq 0 , \dfrac { 5 } { 2 }\\[6pt]$
13.

$\dfrac { x + 3 } { x - 7 } ; x \neq - 2,7\\[6pt]$

15.

$- \dfrac { x + 1 } { 5 x + 6 } ; x \neq - \dfrac { 6 } { 5 } , 1\\[6pt]$
17.

$- \dfrac { 4 x + 3 } { x - 3 } ; x \neq \pm 3\\[6pt]$
19.

$\dfrac { 1 } { x + 4 } ; x \neq 1 , \pm 4\\[6pt]$

B: Multiply or Divide Rational Expressions.

Exercise $\PageIndex{2}$

$\bigstar$ Multiply or divide as indicated, state the restrictions, and simplify.

$\dfrac { 14 ( x + 12 ) ^ { 2 } } { 5 x ^ { 3 } } \cdot \dfrac { 45 x ^ { 4 } } { 2 ( x + 12 ) ^ { 3 } }\\[6pt]$
$\dfrac { 27 x ^ { 6 } } { 20 ( x - 7 ) ^ { 3 } } \cdot \dfrac { ( x - 7 ) ^ { 5 } } { 54 x ^ { 7 } }\\[6pt]$
$\dfrac { x ^ { 2 } - 64 } { 36 x ^ { 4 } } \cdot \dfrac { 12 x ^ { 3 } } { x ^ { 2 } + 4 x - 32 }\\[6pt]$
$\dfrac { 50 x ^ { 5 } } { x ^ { 2 } + 6 x - 27 } \cdot \dfrac { x ^ { 2 } - 81 } { 125 x ^ { 3 } }\\[6pt]$
$\dfrac { 2 x ^ { 2 } + 7 x + 5 } { 3 x ^ { 2 } } \cdot \dfrac { 15 x ^ { 3 } - 30 x ^ { 2 } } { 2 x ^ { 2 } + x - 10 }\\[6pt]$

$\dfrac { 3 x ^ { 2 } + 14 x - 5 } { 2 x ^ { 2 } + 11 x + 5 } \cdot \dfrac { 4 x ^ { 2 } + 4 x + 1 } { 6 x ^ { 2 } + x - 1 }\\[6pt]$
$\dfrac { x ^ { 2 } + 4 x - 21 } { 5 x ^ { 2 } + 10 x } \div \dfrac { x ^ { 2 } - 6 x + 9 } { x ^ { 2 } + 9 x + 14 }\\[6pt]$
$\dfrac { x ^ { 2 } - 49 } { 9 x ^ { 2 } - 24 x + 16 } \div \dfrac { 2 x ^ { 2 } - 13 x - 7 } { 6 x ^ { 2 } - 5 x - 4 }\\[6pt]$
$\dfrac { 5 x ^ { 2 } + x - 6 } { 4 x ^ { 2 } - 7 x - 15 } \div \dfrac { 1 - x ^ { 2 } } { 4 x ^ { 2 } + 9 x + 5 }\\[6pt]$
$\dfrac { 6 x ^ { 2 } - 8 x - 8 } { 4 - 9 x ^ { 2 } } \div \dfrac { 3 x ^ { 2 } - 4 x - 4 } { 9 x ^ { 2 } + 12 x + 4 }\\[6pt]$

$\dfrac { x ^ { 2 } + 4 x - 12 } { x ^ { 2 } - 2 x - 15 } \div \dfrac { 2 x ^ { 2 } - 13 x + 18 } { 6 x ^ { 2 } - 31 x + 5 }\\[6pt]$
$\dfrac { 8 x ^ { 2 } + x - 9 } { 25 x ^ { 2 } - 1 } \div \dfrac { 2 x ^ { 2 } - x - 1 } { 10 x ^ { 2 } - 3 x - 1 }\\[6pt]$
$\dfrac { 1 } { 12 a b } \cdot \dfrac { 50 a ^ { 2 } ( a - b ) ^ { 2 } } { a ^ { 2 } - b ^ { 2 } } \cdot \dfrac { 6 b } { a ( a - b ) }\\[6pt]$
$\dfrac { b ^ { 2 } - a ^ { 2 } } { ( a - b ) ^ { 2 } } \cdot \dfrac { 12 a ( a - b ) } { 36 a ^ { 2 } b } \cdot \dfrac { 9 a b ( a - b ) } { a + b }\\[6pt]$
$\dfrac { x ^ { 3 } + y ^ { 3 } } { 5 x y } \cdot \dfrac { x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } - 2 x y + y ^ { 2 } } \cdot \dfrac { 25 x ^ { 2 } y } { ( y + x ) ^ { 2 } }\\[6pt]$

Answers to odd exercises:

21.

$\dfrac { 63 x } { x + 12 } ; x \neq - 12,0\\[6pt]$
23.

$\dfrac { x - 8 } { 3 x ( x - 4 ) } ; x \neq - 8,0,4\\[6pt]$
25.

$5 ( x + 1 ) ; x \neq - \dfrac { 5 } { 2 } , 0,2$

27.

$\dfrac { ( x + 7 ) ^ { 2 } } { 5 x ( x - 3 ) } ; x \neq - 3 , - 2,0,3\\[6pt]$
29.

$- \dfrac { 5 x + 6 } { x - 3 } ; x \neq - \dfrac { 5 } { 4 } , - 1,1,3\\[6pt]$
31.

$\dfrac { ( x + 6 ) ( 6 x - 1 ) } { ( x + 3 ) ( 2 x - 9 ) } ; x \neq - 3 , \dfrac { 1 } { 6 } , 2 , \dfrac { 9 } { 2 } , 5$

33.

$\dfrac { 25 } { a + b }\\[6pt]$
35.

$\dfrac { 5 x \left( x ^ { 2 } - x y + y ^ { 2 } \right) } { x - y }$

$\bigstar$ Multiply or divide as indicated, state the restrictions, and simplify.

$\dfrac { 3 x y ^ { 2 } } { ( 2 y + x ) ^ { 2 } } \cdot \dfrac { 2 x ^ { 2 } + 5 x y + 2 y ^ { 2 } } { 9 x ^ { 2 } } \cdot \dfrac { x ^ { 3 } + 8 y ^ { 3 } } { 6 x y ^ { 2 } + 3 y ^ { 3 } }\\[6pt]$
$\dfrac { 2 x + 5 } { x - 3 } \cdot \dfrac { x ^ { 2 } - 9 } { 5 x ^ { 4 } } \div \dfrac { 2 x ^ { 2 } + 15 x + 25 } { 25 x ^ { 5 } }\\[6pt]$
$\dfrac { 5 x ^ { 2 } - 15 x } { 9 x ^ { 2 } - 4 } \cdot \dfrac { 3 x - 2 } { 20 x ^ { 3 } } \div \dfrac { x - 3 } { 3 x ^ { 2 } - x - 2 }\\[6pt]$
$\dfrac { x ^ { 2 } + 5 x - 50 } { x ^ { 2 } + 5 x - 14 } \div \dfrac { x ^ { 2 } - 25 } { x ^ { 2 } - 49 } \cdot \dfrac { x - 2 } { x ^ { 2 } + 3 x - 70 }\\[6pt]$

$\dfrac { x ^ { 2 } - x - 56 } { 4 x ^ { 2 } - 4 x - 3 } \div \dfrac { 2 x ^ { 2 } + 11 x - 21 } { 25 - 9 x ^ { 2 } } \cdot \dfrac { 4 x ^ { 2 } - 12 x + 9 } { 3 x ^ { 2 } - 19 x - 40 }\\[6pt]$
$\dfrac { 20 x ^ { 2 } - 8 x - 1 } { 6 x ^ { 2 } + 13 x + 6 } \div \dfrac { 1 - 100 x ^ { 2 } } { 3 x ^ { 2 } - x - 2 } \cdot \dfrac { 10 x - 1 } { 2 x ^ { 2 } - 3 x + 1 }\\[6pt]$
$\dfrac { 12 x ^ { 2 } - 13 x + 1 } { x ^ { 2 } + 18 x + 81 } \div \left( 144 x ^ { 2 } - 1 \right) \cdot \dfrac { x ^ { 2 } + 14 x + 45 } { 12 x ^ { 2 } - 11 x - 1 }\\[6pt]$

Answers to odd exercises:

37.

$\dfrac { 5 x ( x + 3 ) } { x + 5 }\\[6pt]$

39.

$\dfrac { 1 } { x + 5 }\\[6pt]$

41.

$- \dfrac { 1 } { 2 x + 3 }\\[6pt]$

C: Add or Subtract Rational Expressions.

Exercise $\PageIndex{3}$

$\bigstar$ Add or subtract and simplify. State the restrictions.

$\dfrac { 3 x } { 3 x + 4 } + \dfrac { 2 } { 3 x + 4 }\\[6pt]$
$\dfrac { 3 x } { 2 x - 1 } - \dfrac { 2 x + 1 } { 2 x - 1 }\\[6pt]$
$\dfrac { x - 2 } { 2 x ^ { 2 } - 11 x - 6 } + \dfrac { x + 3 } { 2 x ^ { 2 } - 11 x - 6 }\\[6pt]$
$\dfrac { 4 x - 1 } { 3 x ^ { 2 } + 2 x - 5 } - \dfrac { x - 6 } { 3 x ^ { 2 } + 2 x - 5 }\\[6pt]$
$\dfrac { 1 } { x } - 2 x\\[6pt]$
$\dfrac { 4 } { x ^ { 3 } } - \dfrac { 1 } { x }\\[6pt]$

$\dfrac { 1 } { x - 1 } - 5\\[6pt]$
$\dfrac { 1 } { x + 7 } - 1\\[6pt]$
$\dfrac { 1 } { x - 2 } - \dfrac { 1 } { 3 x + 4 }\\[6pt]$
$\dfrac { 2 } { 5 x - 2 } - \dfrac { x } { x + 3 }\\[6pt]$
$\dfrac { 1 } { x ^ { 2 } } + \dfrac { 1 } { x - 2 }\\[6pt]$
$\dfrac { 2 x } { x } + \dfrac { 2 } { x - 2 }\\[6pt]$

$\dfrac { 3 x - 7 } { x ( x - 7 ) } + \dfrac { 1 } { 7 - x }\\[6pt]$
$\dfrac { 2 } { 8 - x } + \dfrac { 3 x ^ { 2 } - 1 } { x ^ { 2 } ( x - 8 ) }\\[6pt]$
$\dfrac { x - 1 } { x ^ { 2 } - 25 } - \dfrac { 2 } { x ^ { 2 } - 10 x + 25 }\\[6pt]$
$\dfrac { x + 1 } { 2 x ^ { 2 } + 5 x - 3 } - \dfrac { x } { 4 x ^ { 2 } - 1 }\\[6pt]$
$\dfrac { x } { x ^ { 2 } + 4 x } - \dfrac { 2 } { x ^ { 2 } + 8 x + 16 }\\[6pt]$
$\dfrac { 2 x - 1 } { 4 x ^ { 2 } + 8 x - 5 } - \dfrac { 3 } { 4 x ^ { 2 } + 20 x + 25 }\\[6pt]$

Answers to odd exercises:

51.

$\dfrac { 3 x + 2 } { 3 x + 4 } ; x \neq - \dfrac { 4 } { 3 }\\[6pt]$
53.

$\dfrac { 1 } { x - 6 } ; x \neq - \dfrac { 1 } { 2 } , 6\\[6pt]$
55.

$\dfrac { 1 - 2 x ^ { 2 } } { x } ; x \neq 0\\[6pt]$

57.

$\dfrac { -5 x+6 } { x - 1 } ; x \neq 1\\[6pt]$
59.

$\dfrac { 2 ( x + 3 ) } { ( x - 2 ) ( 3 x + 4 ) } ; x \neq - \dfrac { 4 } { 3 } , 2\\[6pt]$
61.

$\dfrac { ( x - 1 ) ( x + 2 ) } { x ^ { 2 } ( x - 2 ) } ; x \neq 0,2$

63.

$\dfrac { 2 x - 7 } { x ( x - 7 ) } ; x \neq 0,7\\[6pt]$
65.

$\dfrac { x ^ { 2 } - 8 x - 5 } { ( x + 5 ) ( x - 5 ) ^ { 2 } } ; x \neq \pm 5\\[6pt]$
67.

$\dfrac { x + 2 } { ( x + 4 ) ^ { 2 } } ; x \neq 0 , - 4\\[6pt]$

$\bigstar$ Add or subtract and simplify. State the restrictions.

$\dfrac { 5 - x } { 7 x + x ^ { 2 } } - \dfrac { x + 2 } { 49 - x ^ { 2 } }\\[6pt]$
$\dfrac { 2 x } { 4 x ^ { 2 } + x } - \dfrac { x + 1 } { 8 x ^ { 2 } + 6 x + 1 }\\[6pt]$
$\dfrac { x - 1 } { 2 x ^ { 2 } - 7 x - 4 } + \dfrac { 2 x - 1 } { x ^ { 2 } - 5 x + 4 }\\[6pt]$
$\dfrac { 2 ( x + 3 ) } { 3 x ^ { 2 } - 5 x - 2 } + \dfrac { 4 - x } { 3 x ^ { 2 } + 10 x + 3 }\\[6pt]$
$\dfrac { x ^ { 2 } } { 4 + 2 x ^ { 2 } } - \dfrac { 2 } { x ^ { 4 } + 2 x ^ { 2 } }\\[6pt]$
$\dfrac { 3 x } { 4 x ^ { 4 } + 6 x ^ { 3 } } - \dfrac { 2 x ^ { 2 } } { 6 x ^ { 3 } + 9 x ^ { 2 } }\\[6pt]$

$\dfrac { 3 x ^ { 2 } - 12 } { x ^ { 4 } - 8 x ^ { 2 } + 16 } - \dfrac { x ^ { 2 } + 2 } { 4 - x ^ { 2 } }\\[6pt]$
$\dfrac { x ^ { 2 } } { 2 x ^ { 2 } + 1 } + \dfrac { 6 x ^ { 2 } - 24 } { 2 x ^ { 4 } - 7 x ^ { 2 } - 4 }\\[6pt]$
$1 + \dfrac { 3 } { x } - \dfrac { 5 x - 1 } { x ^ { 2 } }\\[6pt]$
$4 + \dfrac { 2 } { x } - \dfrac { 6 x - 1 } { x ^ { 2 } }\\[6pt]$
$\dfrac { 2 x } { x - 8 } - \dfrac { 1 } { 3 x + 1 } - \dfrac { 2 x + 9 } { 3 x ^ { 2 } - 23 x - 8 }\\[6pt]$
$\dfrac { 4 x } { x - 2 } - \dfrac { 10 } { 3 x + 1 } - \dfrac { 19 x + 18 } { 3 x ^ { 2 } - 5 x - 2 }\\[6pt]$

$\dfrac { 1 } { x - 1 } + \dfrac { 1 } { ( x - 1 ) ^ { 2 } } - \dfrac { 1 } { x ^ { 2 } - 1 }\\[6pt]$
$\dfrac { 1 } { x - 2 } - \dfrac { 1 } { x ^ { 2 } - 4 } + \dfrac { 1 } { ( x - 2 ) ^ { 2 } }\\[6pt]$
$\dfrac { 2 x + 1 } { x - 1 } - \dfrac { 3 x } { 2 x ^ { 2 } - 3 x + 1 } + \dfrac { x + 1 } { x - 2 x ^ { 2 } }\\[6pt]$
$\dfrac { 5 x ^ { 2 } } { 2 x ^ { 2 } + 2 x } - \dfrac { x ^ { 2 } - 4 x } { x ^ { 2 } - 2 x } + \dfrac { 4 + 2 x ^ { 2 } } { 4 + 2 x - 2 x ^ { 2 } }\\[6pt]$
$\dfrac { x + 2 } { 2x ( 3 x - 2 ) } + \dfrac { 4 x } { ( x - 2 ) ( 3 x - 2 ) } - \dfrac { 3 x + 2 } { 2 x ( x - 2 ) }\\[6pt]$
$\dfrac { 10 x } { x ( x - 5 ) } - \dfrac { 2 x ^ { 2 } } { ( 2 x - 5 ) ( x - 5 ) } - \dfrac { 5 x } { x ( 2 x - 5 ) }\\[6pt]$

Answers to odd exercises:

69.

$\dfrac { 7 ( 5 - 2 x ) } { x ( 7 + x ) ( 7 - x ) } ; x \neq - 7,0,7\\[6pt]$
71.

$\dfrac { x ( 5 x - 2 ) } { ( x - 4 ) ( x - 1 ) ( 2 x + 1 ) } ; \\ x \neq - \dfrac { 1 } { 2 } , 1,4\\[6pt]$
73.

$\dfrac { x ^ { 2 } - 2 } { 2 x ^ { 2 } } ; x \neq 0$

75.

$\dfrac { x ^ { 2 } + 5 } { ( x + 2 ) ( x - 2 ) } ; x \neq \pm 2\\[6pt]$
77.

$\dfrac { ( x - 1 ) ^ { 2 } } { x ^ { 2 } } ; x \neq 0\\[6pt]$
79.

$\dfrac { 2 x - 1 } { x - 8 } ; x \neq - \dfrac { 1 } { 3 } , 8\\[6pt]$

81.

$\dfrac { x ^ { 2 } + 1 } { ( x - 1 ) ^ { 2 } ( x + 1 ) } ; x \neq \pm 1\\[6pt]$
83.

$\dfrac { 2 x + 1 } { x } ; x \neq 0 , \dfrac { 1 } { 2 } , 1\\[6pt]$
85.

$0 ; x \neq 0 , \dfrac { 2 } { 3 } , 2$

$\bigstar$ Add or subtract and simplify.

$x ^ { - 2 } + y ^ { - 2 }\\[6pt]$
$x ^ { - 2 } + ( 2 y ) ^ { - 2 }\\[6pt]$
$2 x ^ { - 1 } + y ^ { - 2 }\\[6pt]$
$x ^ { - 2 } - 4 y ^ { - 1 }\\[6pt]$

$16 x ^ { - 2 } + y ^ { 2 }\\[6pt]$
$x y ^ { - 1 } - y x ^ { - 1 }\\[6pt]$
$3 ( x + y ) ^ { - 1 } + x ^ { - 2 }\\[6pt]$
$2 ( x - y ) ^ { - 2 } - ( x - y ) ^ { - 1 }\\[6pt]$

$a ^ { - 2 } - ( a + b ) ^ { - 1 }\\[6pt]$
$( a - b ) ^ { - 1 } - ( a + b ) ^ { - 1 }\\[6pt]$
$x ^ { - n } + y ^ { - n }\\[6pt]$
$x y ^ { - n } + y x ^ { - n }\\[6pt]$

Answers to odd exercises:

87.

$\dfrac { y ^ { 2 } + x ^ { 2 } } { x ^ { 2 } y ^ { 2 } }\\[6pt]$
89.

$\dfrac { x + 2 y ^ { 2 } } { x y ^ { 2 } }$

91.

$\dfrac { x ^ { 2 } y ^ { 2 } + 16 } { x ^ { 2 } }\\[6pt]$
93.

$\dfrac { 3 x ^ { 2 } + x + y } { x ^ { 2 } ( x + y ) }$

95.

$\dfrac { a + b - a ^ { 2 } } { a ^ { 2 } ( a + b ) }\\[6pt]$
97.

$\dfrac { x ^ { n } + y ^ { n } } { x ^ { n } y ^ { n } }$

D: Simplify Complex Rational Expressions.

Exercise $\PageIndex{4}$

$\bigstar$ Simplify each complex rational expression.

$\dfrac { \dfrac { 75 x ^ { 2 } } { ( x - 3 ) ^ { 2 } } } { \dfrac { 25 x ^ { 3 } } { x - 3 } }\\[6pt]$

$\dfrac { \dfrac { x + 5 } { 36 x ^ { 3 } } } { \dfrac { ( x + 5 ) ^ { 3 } } { 9 x ^ { 2 } } }\\[6pt]$

$\dfrac { \dfrac { x ^ { 2 } - 36 } { 32 x ^ { 5 } } } { \dfrac { x - 6 } { 4 x ^ { 3 } } }\\[6pt]$

$\dfrac { \dfrac { x - 8 } { 56 x ^ { 2 } } } { \dfrac { x ^ { 2 } - 64 } { 7 x ^ { 3 } } }\\[6pt]$

$\dfrac { \dfrac { 5 x + 1 } { 2 x ^ { 2 } + x - 10 } } { \dfrac { 25 x ^ { 2 } + 10 x + 1 } { 4 x ^ { 2 } - 25 } }\\[6pt]$
$\dfrac { \dfrac { 4 x ^ { 2 } - 27 x - 7 } { 4 x ^ { 2 } - 1 } } { \dfrac { x - 7 } { 6 x ^ { 2 } - x - 1 } }\\[6pt]$
$\dfrac { \dfrac { x ^ { 2 } - 4 x - 5 } { 2 x ^ { 2 } + 3 x + 1 } } { \dfrac { x ^ { 2 } - 10 x + 25 } { 2 x ^ { 2 } + 7 x + 3 } }\\[6pt]$
$\dfrac { \dfrac { 5 x ^ { 2 } + 9 x - 2 } { x ^ { 2 } + 4 x + 4 } } { \dfrac { 10 x ^ { 2 } + 3 x - 1 } { 4 x ^ { 2 } + 7 x - 2 } }\\[12pt]$
$\dfrac { x ^ { 2 } } { \dfrac { 1 } { 5 } - \dfrac { 3 } { x } }\\[12pt]$
$\dfrac { \dfrac { 4 } { x } - 3 } { 2 x ^ { 2 } }\\[12pt]$

$\dfrac { \dfrac { 1 } { 3 } - \dfrac { 1 } { x } } { \dfrac { 1 } { 9 } - \dfrac { 1 } { x ^ { 2 } } }\\[6pt]$
$\dfrac { \dfrac { 2 } { 5 } + \dfrac { 1 } { x } } { \dfrac { 4 } { 25 } - \dfrac { 1 } { x ^ { 2 } } }\\[6pt]$
$\dfrac { \dfrac { 1 } { y ^ { 2 } } - 36 } { 6 - \dfrac { 1 } { y } }\\[6pt]$
$\dfrac { \dfrac { 1 } { 5 } - \dfrac { 1 } { y } } { \dfrac { 1 } { y ^ { 2 } } - \dfrac { 1 } { 25 } }\\[6pt]$
$\dfrac { 1 - \dfrac { 6 } { x } + \dfrac { 8 } { x ^ { 2 } } } { 3 - \dfrac { 5 } { x } - \dfrac { 2 } { x ^ { 2 } } }\\[6pt]$
$\dfrac { 2 + \dfrac { 13 } { x } - \dfrac { 7 } { x ^ { 2 } } } { 3 + \dfrac { 1 } { x } - \dfrac { 10 } { x ^ { 2 } } }\\[6pt]$

$\dfrac { 9 - \dfrac { 12 } { x } + \dfrac { 4 } { x ^ { 2 } } } { 9 - \dfrac { 4 } { x ^ { 2 } } }\\[6pt]$
$\dfrac { 4 - \dfrac { 25 } { x ^ { 2 } } } { 4 - \dfrac { 8 } { x } - \dfrac { 5 } { x ^ { 2 } } }\\[6pt]$
$\dfrac { \dfrac { 1 } { x } + \dfrac { 5 } { 3 x - 1 } } { \dfrac { 2 } { 3 x - 1 } - \dfrac { 1 } { x } }\\[6pt]$
$\dfrac { \dfrac { 2 } { x - 5 } - \dfrac { 1 } { x } } { \dfrac { 1 } { x } - \dfrac { 3 } { x - 5 } }\\[6pt]$
$\dfrac { \dfrac { 1 } { x + 1 } + \dfrac { 2 } { x - 2 } } { \dfrac { 2 } { x - 3 } - \dfrac { 1 } { x - 2 } }\\[6pt]$
$\dfrac { \dfrac { 4 } { x + 5 } - \dfrac { 1 } { x - 3 } } { \dfrac { 3 } { x - 3 } + \dfrac { 1 } { 2 x - 1 } }\\[6pt]$

$\dfrac { \dfrac { x - 1 } { 3 x - 1 } - \dfrac { 1 } { x + 1 } } { \dfrac { x - 1 } { x + 1 } - \dfrac { 2 } { x + 1 } }\\[6pt]$
$\dfrac { \dfrac { x + 1 } { 3 x + 5 } - \dfrac { 1 } { x + 3 } } { \dfrac { 2 } { x + 3 } - \dfrac { x + 1 } { x + 3 } }\\[6pt]$
$\dfrac { \dfrac { 2 x + 3 } { 2 x - 3 } + \dfrac { 2 x - 3 } { 2 x + 3 } } { \dfrac { 2 x + 3 } { 2 x - 3 } - \dfrac { 2 x - 3 } { 2 x + 3 } }\\[6pt]$
$\dfrac { \dfrac { x - 1 } { x + 1 } - \dfrac { x + 1 } { x - 1 } } { \dfrac { x + 1 } { x - 1 } - \dfrac { x - 1 } { x + 1 } }\\[6pt]$
$\dfrac { \dfrac { 1 } { 2 x + 5 } - \dfrac { 1 } { 2 x - 5 } + \dfrac { 4 x } { 4 x ^ { 2 } - 25 } } { \dfrac { 1 } { 2 x + 5 } + \dfrac { 1 } { 2 x - 5 } + \dfrac { 4 x } { 4 x ^ { 2 } - 25 } }\\[6pt]$
$\dfrac { \dfrac { 1 } { 3 x - 1 } + \dfrac { 1 } { 3 x + 1 } } { \dfrac { 3 x } { 3 x - 1 } - \dfrac { 1 } { 3 x + 1 } - \dfrac { 6 x } { 9 x ^ { 2 } - 1 } }\\[6pt]$

Answers to odd exercises:

101.

$\dfrac { 3 } { x ( x - 3 ) }\\[6pt]$
103.

$\dfrac { x + 6 } { 8 x ^ { 2 } }\\[6pt]$
105.

$\dfrac { 2 x - 5 } { ( x - 2 ) ( 5 x + 1 ) }\\[6pt]$
107.

$\dfrac { x + 3 } { x - 5 }\\[6pt]$

109.

$\dfrac { 5 x ^ { 3 } } { x - 15 }\\[6pt]$
111.

$\dfrac { 3 x } { x + 3 }\\[6pt]$
113.

$- \dfrac { 6 y + 1 } { y }\\[6pt]$

115.

$\dfrac { x - 4 } { 3 x + 1 }\\[6pt]$
117.

$\dfrac { 3 x - 2 } { 3 x + 2 }\\[6pt]$
119.

$- \dfrac { 8 x - 1 } { x - 1 }\\[6pt]$

121.

$\dfrac { 3 x ( x - 3 ) } { ( x + 1 ) ( x - 1 ) }\\[6pt]$
123.

$\dfrac { x } { ( x + 1 ) ( 3x - 1 ) }\\[6pt]$
125.

$\dfrac { 4 x ^ { 2 } + 9 } { 12 x }\\[6pt]$
127.

$\dfrac { 2 x - 5 } { 4 x }\\[6pt]$

$\bigstar$ Simplify each complex rational expression.

$\dfrac { 1 } { 1 + \dfrac { 1 } { 1 + \dfrac { 1 } { x } } }\\[6pt]$
$\dfrac { \dfrac { 1 } { x } } { 1 - \dfrac { 1 } { 1 + \dfrac { 1 } { x } } }\\[6pt]$
$\dfrac { \dfrac { 1 } { y } - \dfrac { 1 } { x } } { \dfrac { 1 } { y ^ { 2 } } - \dfrac { 1 } { x ^ { 2 } } }\\[6pt]$
$\dfrac { \dfrac { 2 } { y } + \dfrac { 1 } { x } } { \dfrac { 4 } { y ^ { 2 } } - \dfrac { 1 } { x ^ { 2 } } }\\[6pt]$

$\dfrac { \dfrac { 1 } { 25 y ^ { 2 } } - \dfrac { 1 } { x ^ { 2 } } } { \dfrac { 1 } { x } - \dfrac { 1 } { 5 y } }\\[6pt]$
$\dfrac { 16 y ^ { 2 } - \dfrac { 1 } { x ^ { 2 } } } { \dfrac { 1 } { x } - 4 y }\\[6pt]$
$\dfrac { \dfrac { 1 } { b } + \dfrac { 1 } { a } } { \dfrac { 1 } { b ^ { 3 } } + \dfrac { 1 } { a ^ { 3 } } }\\[6pt]$
$\dfrac { \dfrac { 1 } { a } - \dfrac { 1 } { b } } { \dfrac { 1 } { b ^ { 3 } } - \dfrac { 1 } { a ^ { 3 } } }\\[6pt]$

$\dfrac { \dfrac { x } { y } - \dfrac { y } { x } } { \dfrac { 1 } { y ^ { 2 } } - \dfrac { 2 } { x y } + \dfrac { 1 } { x ^ { 2 } } }\\[6pt]$
$\dfrac { \dfrac { 2 } { y } - \dfrac { 5 } { x } } { 4 x - \dfrac { 25 y ^ { 2 } } { x } }\\[6pt]$
$\dfrac { x ^ { - 1 } + y ^ { - 1 } } { y ^ { - 2 } - x ^ { - 2 } }\\[6pt]$
$\dfrac { y ^ { - 2 } - 25 x ^ { - 2 } } { 5 x ^ { - 1 } - y ^ { - 1 } }\\[6pt]$
$\dfrac { 1 - x ^ { - 1 } } { x - x ^ { - 1 } }\\[6pt]$

$\dfrac { 16 - x ^ { - 2 } } { x ^ { - 1 } - 4 }\\[8pt]$
$\dfrac { 1 - 4 x ^ { - 1 } - 21 x ^ { - 2 } } { 1 - 2 x ^ { - 1 } - 15 x ^ { - 2 } }\\[8pt]$
$\dfrac { x ^ { - 1 } - 4 \left( 3 x ^ { 2 } \right) ^ { - 1 } } { 3 - 8 x ^ { - 1 } + 16 \left( 3 x ^ { 2 } \right) ^ { - 1 } }\\[8pt]$
$\dfrac { ( x - 3 ) ^ { - 1 } + 2 x ^ { - 1 } } { x ^ { - 1 } - 3 ( x - 3 ) ^ { - 1 } }\\[8pt]$
$\dfrac { ( 4 x - 5 ) ^ { - 1 } + x ^ { - 2 } } { x ^ { - 2 } + ( 3 x - 10 ) ^ { - 1 } }\\[6pt]$

Answers to odd exercises:

129.

$\dfrac { x + 1 } { 2 x + 1 }\\[6pt]$
131.

$\dfrac { x y } { x + y }\\[6pt]$

133.

$- \dfrac { x + 5 y } { 5 x y }\\[6pt]$
135.

$\dfrac { a ^ { 2 } b ^ { 2 } } { a ^ { 2 } - a b + b ^ { 2 } }\\[6pt]$

137.

$\dfrac { x y ( x + y ) } { x - y }\\[6pt]$
139.

$\dfrac { x y } { x - y }\\[6pt]$
141.

$\dfrac { 1 } { x + 1 }\\[6pt]$

143.

$\dfrac { x - 7 } { x - 5 }\\[6pt]$
145.

$- \dfrac { 3 ( x - 2 ) } { 2 x + 3 }\\[6pt]$

E: Mixed Practice with Rational Expressions.

Exercise $\PageIndex{5}$

$\bigstar$ Perform the indicated operations. Simplify the result if possible. State the restrictions.

$\dfrac { 108 x ^ { 3 } } { 12 x ^ { 2 } }\\[3pt]$
$\dfrac { 56 x ^ { 2 } ( x - 2 ) ^ { 2 } } { 8 x ( x - 2 ) ^ { 3 } }\\[6pt]$
$\dfrac { 64 - x ^ { 2 } } { 2 x ^ { 2 } - 15 x - 8 }\\[6pt]$
$\dfrac { 3 x ^ { 2 } + 28 x + 9 } { 81 - x ^ { 2 } }\\[6pt]$
$\dfrac { x ^ { 2 } - 25 } { 5 x ^ { 2 } } \cdot \dfrac { 10 x ^ { 2 } - 15 x } { 2 x ^ { 2 } + 7 x - 15 }\\[6pt]$
$\dfrac { 7 x ^ { 2 } - 41 x - 6 } { ( x - 7 ) ^ { 2 } } \cdot \dfrac { 49 - x ^ { 2 } } { x ^ { 2 } + x - 42 }\\[6pt]$
$\dfrac { 28 x ^ { 2 } ( 2 x - 3 ) } { 4 x ^ { 2 } - 9 } \div \dfrac { 7 x } { 4 x ^ { 2 } - 12 x + 9 }\\[6pt]$
$\dfrac { x ^ { 2 } - 10 x + 24 } { x ^ { 2 } - 8 x + 16 } \div \dfrac { 2 x ^ { 2 } - 13 x + 6 } { x ^ { 2 } + 2 x - 24 }\\[6pt]$
$\dfrac { 5 x - 6 } { x ^ { 2 } - 36 } - \dfrac { 4 x } { x ^ { 2 } - 36 }\\[6pt]$
$\dfrac { 2 } { x } + 5 x\\[6pt]$

$\dfrac { 5 } { x - 5 } + \dfrac { 1 } { 2 x }\\[6pt]$
$\dfrac { x } { x - 2 } + \dfrac { 3 } { x + 3 }\\[6pt]$
$\dfrac { 7 ( x - 1 ) } { 4 x ^ { 2 } - 17 x + 15 } - \dfrac { 2 } { x - 3 }\\[6pt]$
$\dfrac { 5 } { x } - \dfrac { 19 x + 25 } { 2 x ^ { 2 } + 5 x }\\[6pt]$
$\dfrac { x } { x - 5 } - \dfrac { 2 } { x - 3 } - \dfrac { 5 ( x - 3 ) } { x ^ { 2 } - 8 x + 15 }\\[6pt]$
$\dfrac { 3 x } { 2 x - 1 } - \dfrac { x - 4 } { x + 4 } + \dfrac { 12 ( 2 - x ) } { 2 x ^ { 2 } + 7 x - 4 }\\[6pt]$
$\dfrac { 1 } { t - 1 } + \dfrac { 1 } { ( t - 1 ) ^ { 2 } } - \dfrac { 1 } { t ^ { 2 } - 1 }\\[6pt]$
$\dfrac { 1 } { t - 1 } - \dfrac { 2 t - 5 } { t ^ { 2 } - 2 t + 1 } - \dfrac { 5 t ^ { 2 } - 3 t - 2 } { ( t - 1 ) ^ { 3 } }\\[6pt]$
$2 x ^ { - 1 } + x ^ { - 2 }\\[6pt]$
$( x - 4 ) ^ { - 1 } - 2 x ^ { - 2 }\\[6pt]$

$\dfrac { \dfrac { 1 } { 7 } + \dfrac { 1 } { x } } { \dfrac { 1 } { 49 } - \dfrac { 1 } { x ^ { 2 } } }\\[6pt]$
$\dfrac { \dfrac { 1 } { 100 } - \dfrac { 1 } { x ^ { 2 } } } { \dfrac { 1 } { x } - \dfrac { 1 } { 10 } }\\[6pt]$
$\dfrac { \dfrac { 3 } { x } - \dfrac { 1 } { x - 5 } } { \dfrac { 5 } { x + 2 } - \dfrac { 2 } { x } }\\[6pt]$
$\dfrac { 1 - \dfrac { 12 } { x } + \dfrac { 35 } { x ^ { 2 } } } { 1 - \dfrac { 25 } { x ^ { 2 } } }\\[6pt]$
$\dfrac { x - 4 x ^ { - 1 } } { 2 - 5 x ^ { - 1 } + 2 x ^ { - 2 } }\\[6pt]$
$\dfrac { 8 x ^ { - 1 } + y ^ { - 1 } } { y ^ { - 2 } - 64 x ^ { - 2 } }\\[6pt]$

Answers to odd exercises:

151.

$9 x ; x \neq 0\\[6pt]$
153.

$- \dfrac { x + 8 } { 2 x + 1 } ; x \neq - \dfrac { 1 } { 2 } , 8\\[6pt]$
155.

$\dfrac { x - 5 } { x } ; x \neq - 5,0 , \dfrac { 3 } { 2 }\\[6pt]$
157.

$\dfrac { 4 x ( 2 x - 3 ) ^ { 2 } } { 2 x + 3 } ; x \neq \pm \dfrac { 3 } { 2 } , 0\\[6pt]$

159.

$\dfrac { 1 } { x + 6 } ; x \neq \pm 6\\[6pt]$
161.

$\dfrac { 11 x - 5 } { 2 x ( x - 5 ) } ; x \neq 0,5\\[6pt]$
163.

$- \dfrac { 1 } { 4 x - 5 } ; x \neq \dfrac { 5 } { 4 } , 3\\[6pt]$

165.

$\dfrac { x - 5 } { x - 3 } ; x \neq 3,5\\[6pt]$
167.

$\dfrac { t ^ { 2 } + 1 } { ( t + 1 ) ( t - 1 ) ^ { 2 } } ; t \neq \pm 1\\[6pt]$
169.

$\dfrac { 2 x + 1 } { x ^ { 2 } } ; x \neq 0\\[6pt]$

171.

$\dfrac { 7 x } { x - 7 }; \;x \ne 0, 7, -7\\[6pt]$
173.

$\dfrac { ( x + 2 ) ( 2 x - 15 ) } { ( x - 5 ) ( 3 x - 4 ) }; \\ \;x \neq -2, 0, 5, \frac{4}{3}\\[6pt]$
175.

$\dfrac { x ( x + 2 ) } { 2 x - 1 } \; d\neq 0, 2, \frac{1}{2} \\[6pt]$

$\star$

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A: Simplify Rational Expressions.

B: Multiply or Divide Rational Expressions.

C: Add or Subtract Rational Expressions.

D: Simplify Complex Rational Expressions.

E: Mixed Practice with Rational Expressions.

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